Lest anyone mistake me, I am not saying one side is bad or good. The two sides are neither bad nor good, just two sides of the same coin. Thus the inversions are just the other side of the coin too...the other side of the cycle. There is an ouroboros, but like PrettyPig pointed out wheels can turn in either direction and IMO it is currently running in reverse undoing the harm the Children believe their three swords caused. That's not to say that there couldn't be two factions of Children. The remaining living Children may be the ones trying to reverse the damage, while the dead godhead may be the "wroth" side.
I just want to point out that geographically the Iron Islands and Dragonstone are physically on opposite sides of Westeros from each other, just as Casterly Rock and Dorne. Surely the readers of this thread have noticed the parallel inversions to the two sides? What happened in the west of Casterly Rock is now happening in reverse in the east in Dorne, and what happened in the east with the Blackfyres and Targaryens are now happening inversely to the Greyjoys in the west. What are the explanations for this and why does it seem like the Drowned God is north of the Wall? I've tried to answer these questions with the parallel inversion theory, but are there other explanations?

I know, LmL...and I appreciate it. I was just trying to gently steer this ship.
As a maester, Aemon cared for his "drowned men" of the Nights Watch. He was their moral compass and they would strain to hear his whisper soft voice when he gave council. I had commented that he would have encountered the men in the tunnels touching an arm or hand in passing just like Damphair in a crowd.
And of course Patchface is a drowned prophet currently at the Wall preaching his Jabberwocky.

Asha versus Rhaella - Part 4 (last one!)
There is much talk about the kingsmoot, which Asha initially thinks, This is something new…or rather, something very old. The Targaryens had their own “kingsmoots”. Whenever there wasn’t a clear heir a Great Council was called. More than once a Great Council would pass over a female Targaryen heir in favor of a younger son just like the Ironborn are doing to Asha. Rodrik sighs, “You will not want to hear this, Asha, but you will not be chosen. No woman has ever ruled the ironborn.”
Rodrik prays the kingsmoot is not bloody since the last time the Ironborn met four thousand years ago for a kingsmoot, Nagga’s ribs turned red with the gore. Likewise, each Great Council the Targaryens called resulted in bloodshed.
Rodrik tells Asha that a kingsmoot and a dream of kingship is a madness in their blood. It’s land they need, not crowns. This line of thinking struck me as being similar to when Aegon the Conqueror came to Westeros. His family had the small island of Dragonstone after the Doom, and it may have been that more land was needed, but the dream of kingship was a madness in his blood and he took Westeros with his dragons instead of working peacefully in order to improve the lot of his people. Pyke is very much like Dragonstone. Both are small stony outcroppings that don’t provide enough resources for their people. Rodrik urges Asha to make common cause with either Stannis or Tywin to improve the lot of the Ironborn. If they help one of them win then they can claim the land that they need from a grateful king.
Rodrik councils Asha that the Old Way of reaving by ship served them well when the isles were but one small kingdom amongst many, but Aegon’s Conquest put an end to that. What is ironic is that the same could be said for Aegon. Having dragons served them well when Valyria was full of dragonlords, but after all the other families perished in the Doom he abused his powerful dragons to take the Seven Kingdoms for his own by conquest rather than by peaceful negotiation.

Asha versus Rhaella - Part 3
When Asha was looking for her uncle Rodrik, she turned to Three-Tooth, an old woman who had been her uncle’s steward since she was known as Twelve-Tooth, or for as long as Asha could remember. The woman was so old that a septon once quipped that she must have nursed the Crone. When Asha arrived to Ten Towers, she sailed in on her ship, Black Wind with captives Lady Sybelle Glover of Deepwood Motte with her son Gawen and infant daughter Erena. She was concerned for Lady Glover’s comfort and for the well being of her children. She is particularly worried about the health of the infant and tells Three-Tooth that if the babe were to die, no one will be sorrier than you. To me this echoes both an infant Rhaegar born at Summerhall, and his and Elia’s son Aegon.
Rhaella’s uncle, Prince Duncan married Jenny of Oldstones who brought an old wood’s witch to court. This old woman prophesied that the prince that was promised would be born from the line of Aerys and Rhaella. When King Jaehaerys II heard the prophecy he arranged the marriage between his two children.
Asha’s Lady mother Alannys, and aunt Lady Gwynesse have both settled themselves into the Widow’s Tower at Ten Towers. Lady Alannys is described as frail and sickly, and Lady Gwynesse continues to mourn for a husband long dead and walks around muttering how Ten Towers should be hers because she is older than her brother Rodrik.
At the Red Keep, Rhaella would have been near two similar women as Asha. Princess Elia of Dorne’s quarters were in Maegor’s Holdfast and she has been described as frail and sickly, while Cersei was angry that she didn't get to marry Rhaegar, and would have been walking around muttering that she should be Tywin’s heir since she was older than Jaime.
Lady Alannys grieved for two sons, Rodrik and Maron who were killed during Balon’s uprising against King Robert. Princess Elia gave birth to two children, though the birth of Rhaenys left her bed ridden for half a year, and she nearly died giving birth to Aegon. Both children were killed when Robert’s forces sacked Kings Landing. If Lady Alannys is more like Elia, then Lady Glover is someone else. Possible candiates are Ashara or Lyanna.

Asha versus Rhaella - Part 2
At the kingsmoot Asha named all the northern holdfasts that the Ironborn held: Moat Cailin, Deepwood Motte, Torrhen’s Square, even Winterfell, but she asks what do we have to show for it? She calls her supporters her Black Wind men. This name may symbolic of Balerion the Black Dread, one of the three great dragons of Aegon the Conqueror. Asha’s men dump out the “riches” of the north: pebbles from the Stony Shore, pinecones from Deepwood Motte, and turnips from Winterfell. Asha’s point is that political peace is more valuable than plunder. Many men cheered for Asha, but they were shouted down by, “No craven’s peace!” and then calls for Victarion. I wonder if the three chests Asha empties represent the three houses of Arryns, Manderlys, and Starks that supported Rhaenyra’s claim for the Iron Throne? These houses were also on the same side again for Robert’s Rebellion.
Rhaegar gathered supporters under the guise of a tourney at Harrenhal, and as readers we assumed that he was gathering support for himself, but what if he meant to support his mother, Rhaella? Certainly she would have the more lawful claim should Aerys die.
Asha waited at Ten Towers to see which men will answer her uncle’s call to come and support her claim to the Seastone Chair. “Too few,” she thinks, “too few by far.”
The Red Keep is the castle in Kings Landing and is home to the royal family. It has seven massive drum-towers with iron ramparts, and three inner towers: Maegor’s Holdfast, Tower of the Hand, and the White Sword Tower.
Rodrik, nicknamed the Reader, is Lord of Harlaw and Asha’s favorite uncle. Rodrik reads so much that he has requested a Myrish lense to assist with his vision. He keeps septons at Ten Towers just to take care of his books. In this chapter he is reading Archmaester Marwyn’s Book of Lost Books.
Rhaella’s eldest and favorite son, Rhaegar read obsessively as a child, to the point that jests were made about him reading with a candle in Rhaella’s womb. He became a noted fighter because of something that he had read in an old book had motivated him to become a warrior.

I'd really like to try to keep the discussion on parallel inversions.
I posted some parallel inversions between the followers of the Drowned God and the Nights Watch, and I'm also going to drop some parallels between Asha Greyjoy and Rhaella Targaryen.
Asha versus Rhaella - Part 1
When Damphair told Balon that Theon was a weakling, Balon said he prayed that the wolves would kill him so that he wouldn’t stand in Asha’s way. Damphair thought Balon was blind if he believed Asha could succeed him. Aeron tried to tell him that no woman will ever rule the ironborn, not even a woman such as Asha, but Balon did not wish to hear.
When the death of King Aerys I left no clear heir, Lord Brynden Rivers as Hand of the King called for a Great Council to decide who would inherit the Iron Throne. The merits of each living Targaryen were discussed. There have been several Great Councils during the Targaryen reign, but a woman has never been selected.
After the death of King Balon, Damphair believed Victarion should rule though Euron was the elder brother, and even though Theon and Asha, having the more lawful claim, would come before them both. Aeron dismisses both children of Balon as unfit, and claims Euron cannot be king because he’s godless.
Aenys Blackfyre petitioned the Great Council to peacefully be considered has heir to the throne. Bloodraven offered Aenys safe conduct to Kings Landing, but once he arrived he was arrested and beheaded. Euron is equally skilled as Bloodraven. He’s cunning, shrewd, and ruthless. Aenys Blackfyre’s parallel is Baelor Blacktyde. Baelor was an Asha supporter at the kingsmoot. He refused to acknowledge Euron and attempted to leave, but Victarion blocked his way and Baelor is delivered to King Euron in chains. As punishment for his refusal to recognize his new liege, he is cut into seven pieces representing his faith to the Seven by Euron’s crew. Euron claims Baelor’s sable cloak for himself.
The Reader (Rodrik Harlaw) had sent ravens summoning all of Asha’s friends to Harlaw, because he believed that Balon meant for her to sit the Seastone Chair. He did this, even though privately he told Asha that the Ironborn would never select a woman.
The Great Council was responsible for choosing the next heir to the Iron Throne. Asha parallels, not only Rhaella, but she could equally be reliving Rhaenyra Targaryen, first-born child of King Viserys I who was passed over by her younger half-brother Aegon II sparking the civil war known as the Dance of the Dragons. Speaking of dragons, Asha’s ship Black Wind brings to mind Aegon’s dragon, Balerion the Black Dread, and Balerion does sound a little like Balon.

Drowned Men versus Nights Watch
In the Iron Islands a “drowned man” is a convert to the faith of the Drowned God. Their baptism is a literal drowning. They are then resuscitated back to life with something close to CPR, so that they can rise again stronger and harder. In The Drowned Man AFFC chpt 19 Damphair made his way out of the water and across the strand when a drowned man returning from the call of nature stumbled into him in the darkness. “Damphair,” he murmured. Aeron laid a hand upon his head and blessed him. IMO Aemon Targaryen is a parallel inversion to Aeron Greyjoy. I can imagine a blind Aemon Targaryen making his way along the Wall and stumbling into another man of the Nights Watch. The area north of the Wall symbolically is a great northern sea. Patchface says that the north is upside down so everybody at the Wall and north of it is underwater effectively making the black brothers of the Nights Watch “drowned men”.
Nagga’s ribs are situated on the crown of a hill, forty-four monstrous stone ribs rising up like the trunks of great trees. It’s said she had fed on krakens and leviathans and drowned whole islands in her wrath, yet the Grey King had slain her and the Drowned God had changed her bones to stone.
What could be Nagga’s parallel in Aemon’s time? Certainly the Wall is wondrous to behold, and monstrous. The kraken is the sigil of the Greyjoys, and the dragon is the sigil of the Targaryens, but what is a leviathan? The Bible describes leviathans as great sea monsters. Ancient eastern origins describe leviathans as a seven headed serpent. Euron cut Baelor Blacktyde into seven pieces symbolizing his intent to tear Westeros apart. The inversion to this would be the Targaryen’s unification of the Seven Kingdoms by adopting the Faith of the Seven. There’s also the possibility that the leviathan symbolism is the growing danger coming from the northern sea. The Drowned God would certainly fit as a parallel to the old gods of the Children since they were forced north when the Targaryens adopted the Faith of the Seven.
The Grey King that turned Nagga’s ribs to stone sounds similar to the Grey Lord who was also Prince Garin the Great, a historical figure of the Rhoynar who beseeched Mother Rhoyne to curse the men of Volantis and Valyria with a damp fog that causes greyscale. Greyscale eventually consumes it’s victim turning the flesh to stone. The Grey King of the Greyjoys turned Nagga’s ribs to stone. So they have “turning things into stone” in common.
In the great northern sea the dead "drowned men" rise as wights when it is dark and the cold wind is rising. There is no name for wightification and no lord or king is blamed, but the Grey King could also be said to parallel the Nights King. Aeron recalled that it was at Nagga’s ribs where the Grey King took his mermaid wife and planned his wars against the Storm God, whereas the Nights King spied his Nights Queen from atop the Wall, and together they ensorcelling the Watch, bent them to his will, and sacrificed children to the Others.
Damphair tells his followers: “We were born from the sea, and to the sea we all return. The Storm God in his wrath plucked Balon from his castle and cast him down, yet now he feasts beneath the waves in the Drowned God’s watery halls.”….”Yet what is dead may never die, but rises again, harder and stronger!”
Once a man of the Nights Watch takes his vows, he cannot leave. Their vows are said either in a sept or before a hearttree: “Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post….”

A nice parallel inversion I would say....I think you could draw some conclusions regarding the process by wondering what the inversion would be? The ice priestess could remain human and non-magical like Gilly since that would be more of an inversion to a warm magical fire priestess. I like FFR's version that Craster's children are actually brought back to the cave and sacrificed. Nothing magical about that part. The transformation then is the use of the blood to feed the weirwoods into producing the white walker out of water and cold air.

I think you are on to something here. After all when Bran learned of the Childen's history he said that if it happened to humans that they would be "wroth" and want to kill their enemy. Do you suspect a division amongst the Children and/or the godhead?
The blood of Craster's children was likely the sacrifice and then the rest would be viewed as just meat. Makes sense to me!

I think the best way to demonstrate my theory would be to get into some examples. I'd like to build on the Mormonts, but in order to get there I have to do what I call the "befores". You know, before I paint the wall it'll need to be primed, but before I prime it I should go around and calk nail holes and other imperfections, etc. Sometimes the "befores" can stretch back so much that you end up procrastinating because there are too many things that need to be done first. So, before I get to the Mormonts I need to return to the Greyjoys.
I touched briefly how I think Victarion is mirroring Bittersteel and Euron is Bloodraven. Building on that, who would be possible parallel inversions to Balon and Asha then? While Balon is both Victarion and Euron's brother IMO since he's a king he more properly mirror's Bittersteel and Bloodraven's father, King Aegon IV, while his daughter Asha is a princess and seems to more properly mirror Rhaella Targaryen.
One of the Lords that came to Asha’s support was an old boyfriend, Trisfifer Botley who apparantly still held a torch for her. Tristifer was fostered by her mother after Theon was taken as Ned Stark’s ward, and being they were both young and hormonal they experimented, fumbling around under each other’s clothes and making out. This should be awkward, Asha thinks because apparently Asha doesn’t return Tristifer’s feelings and was actually relieved when they were caught by a maester causing Tristifer to be sent away. Tristifer has apparently been saving himself for Asha ever since, never touching another women.
Ser Barristan Selmy claimed that in her youth, Rhaella was in love with Ser Bonifer Hasty, a young landed knight from the stormlands, who once named her queen of love and beauty. It was a brief fling, as Ser Bonifer was of too low birth to ever be a serious suitor for a Princess. When Rhaella married Aerys, Bonifer devoted himself to the Faith of the Seven.
More to come...

There isn't any reason why they cannot be both POV transformation chapters and inversion chapters. I have links to the chapters that I've completed in my signature. I'm only up to The Reaver chapter, so only a quarter of the way through but the eight that I have done are full of parallel inversions. I didn't include them in the OP because I was trying to lay the foundation upon which to build the theory. How can someone new to the idea have any idea where I'm coming from if I simply jumped into one of the chapters without explanation?
I had suggested a thread about tracing Arya and Sansa's routes, but I think what I'd rather do is a thread for each chapter here on the W to discuss whether my theory proves true.
There is a point to breaking the wheel and sending it in reverse, because it'll return Westeros to the time where the Children think they've gone wrong by reversing the three swords. It'll lead to their own extinction, but like Varys they will do it for the good of the realm.

Thank you for sharing your echoes of the fight at the tower of joy. I still have the opinion that all of them may be echoes of an original fight not mentioned in the story, and connected to the forging of the third sword that led to the breaking of the world.
I understand that there are echoes and inversions outside the titled chapters. There are many fascinating examples that others have brought into this thread, but I still believe the titled chapters are meant to be significant. They are the Jabberwocky for us to decipher. IMO the echoes, parallels, and inversions early on in the first three books were our education...we were being instructed on how to learn the "song". Pain Killer Jane said this upthread:
...also called dreaming track, is one of the paths across the land (or sometimes the sky) which mark the route followed by localised 'creator-beings' during the Dreamtime......
A knowledgeable person is able to navigate across the land by repeating the words of the song, which describe the location of landmarks, waterholes, and other natural phenomena. In some cases, the paths of the creator-beings are said to be evident from their marks, or petrosomatoglyphs, on the land, such as large depressions in the land which are said to be their footprints.
By singing the songs in the appropriate sequence, indigenous people could navigate vast distances, often travelling through the deserts of Australia's interior.
Since a songline can span the lands of several different language groups, different parts of the song are said to be in those different languages. Languages are not a barrier because the melodic contour of the song describes the nature of the land over which the song passes. The rhythm is what is crucial to understanding the song. Listening to the song of the land is the same as walking on this songline and observing the land.
What Pain Killer Jane posted about using a song to navigate isn't anything I've read before, but it is a clarification of exactly what has been rattling and bouncing around in my brain all these months that I have been working on this theory. When I read this I went "YES!" This is what I've been thinking! I just didn't know how to explain it properly.
The first three books were our primers...the books that taught us "the song"...the inversion song, so when we got to A Feast For Crows chapter one The Prophet we were prepared to navigate the land by repeating the words of the song. The rhythm is what is crucial to understanding the song. I'm not quite fluent, but I think I'm picking up on it.